There is a controversy of how to interpret interactions of electrons with a large spatial coherence with light and matter. When such an electron emits a photon, it can do so either as if its charge were confined to a point within a coherence length, the region where a square modulus of a wave function ${\left|\psi \right|}^2$ is localized, or as a continuous cloud of space charge spread over it. This problem was addressed in a recent study [R. Remez et al., Phys. Rev. Lett. 123, 060401 (2019)] where a conclusion was drawn in favor of the first (point) interpretation. Here we argue that there is an alternative explanation for the measurements reported in that paper, which relies on a purely classical concept of a so-called prewave zone and does not allow one to refute the second interpretation. We propose an experiment of Smith-Purcell radiation from a nonrelativistic vortex electron carrying orbital angular momentum, which can unambiguously lead to the opposite conclusion. Beyond the paraxial approximation, the vortex packet has a nonpoint electric quadrupole moment, which grows as the packet spreads and results in a nonlinear $L^3$ growth of the radiation intensity with the length $L$ of the grating when $L$ is much larger than the packet's Rayleigh length. Such a nonlinear effect has never been observed for single electrons and, if detected, it would be a hallmark of the nonpoint nature of charge in a wave packet. Thus, two views on ${\left|\psi \right|}^2$ are complementary to each other and an electron radiates either as a point charge or as a continuous charge flow depending on the experimental conditions and on its quantum state. Our conclusions hold for a large class of non-Gaussian packets and emission processes for which the radiation formation length can exceed the Rayleigh length, such as Cherenkov radiation, transition radiation, diffraction radiation, and so forth.

• Received 7 September 2020
• Accepted 23 December 2020

DOI:https://doi.org/10.1103/PhysRevA.103.012214